Orbital stability


In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form is said to be orbitally stable if any solution with the initial data sufficiently close to forever remains in a given small neighborhood of the trajectory of.

Formal definition

Formal definition is as follows.
Consider the dynamical system
with a Banach space over,
and.
We assume that the system is
-invariant,
so that
for any
and any.
Assume that,
so that is a solution to the dynamical system.
We call such solution a solitary wave.
We say that the solitary wave
is orbitally stable if for any there is
such that for any with
there is a solution defined for all
such that,
and such that this solution satisfies

Example

According to
the solitary wave solution
to the nonlinear Schrödinger equation
where is a smooth real-valued function,
is orbitally stable if the Vakhitov-Kolokolov stability criterion is satisfied:
where
is the charge of the solution,
which is conserved in time.
It was also shown,
that if at a particular value of,
then the solitary wave
is Lyapunov stable, with the Lyapunov function
given by,
where
is the energy of a solution,
with the antiderivative of,
as long as the constant
is chosen sufficiently large.